I'd guess if your bigger inverter has an alarm setpoint of say, 10.5 V it will be better but if it has the same 11V setpoint, it will function the same. The draw from the battery is what's bringing your voltages down, not the function of the inverter.
I'd assume from your statement about the laptop charger causing an alarm that you have a small MSW inverter. By all means, upgrade to a bigger PSW inverter. I recommend the Xantrex Prowatt 1000W inverter. It is silent, durable, and comparatively inexpensive.
